• Corpus ID: 2330752

Circle Packing : A Mathematical Tale

  title={Circle Packing : A Mathematical Tale},
  author={Kenneth Stephenson},
1376 NOTICES OF THE AMS VOLUME 50, NUMBER 11 T he circle is arguably the most studied object in mathematics, yet I am here to tell the tale of circle packing, a topic which is likely to be new to most readers. These packings are configurations of circles satisfying preassigned patterns of tangency, and we will be concerned here with their creation, manipulation, and interpretation. Lest we get off on the wrong foot, I should caution that this is NOT twodimensional “sphere” packing: rather than… 

Patterns Formed by Coins

This article is a gentle introduction to the mathematical area known as circle packing, the study of the kinds of patterns that can be formed by configurations of non-overlapping circles. The first

Circle Patterns on Singular Surfaces

The possible intersection angles and singular curvatures of those circle patterns on Euclidean or hyperbolic surfaces with cone singularities ofHyperideal circle patterns are described, related to results on the dihedral angles of ideal or hyperidealhyperbolic polyhedra.

On the geometry of uniform meandric systems

A meandric system of size n is the set of loops formed from two arc diagrams (non-crossing perfect matchings) on {1, . . . , 2n}, one drawn above the real line and the other below the real line. A

Conformal geometry of simplicial surfaces

What information about a surface is encoded by angles, but not lengths? This question encapsulates the basic viewpoint of conformal geometry, which studies holomorphic or (loosely speaking) angleand

Introduction to circle packing: the theory of discrete analytic functions, by K. Stephenson. Pp. 356. £35.00. 2005. ISBN 0 521 82356 0 (Cambridge University Press).

  • N. Lord
  • Mathematics
    The Mathematical Gazette
  • 2006
equations by radicals and the computation of the Galois group. The last section of this particularly useful chapter is on Hilbert's irreducible theorem, which may be described as a result on the

Some criteria for circle packing types and combinatorial Gauss-Bonnet theorem

We investigate criteria for circle packing(CP) types of disk triangulation graphs embedded into simply connected domains in $ \mathbb{C}$. In particular, by studying combinatorial curvature and the

Irreducible Apollonian Configurations and Packings

The main result is to show how to find a small field that can realize irreducible Apollonian configurations and also give a method to relate the bends of the new circles to the bending of the circles forming the curvilinear triangle.

The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity

We prove that the Tutte embeddings (a.k.a. harmonic/embeddings) of certain random planar maps converge to $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Specifically, we treat mated-CRT maps,

Title Irreducible Apollonian Configurations and Packings Permalink

An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. One

Mating of trees for random planar maps and Liouville quantum gravity: a survey

We survey the theory and applications of mating-of-trees bijections for random planar maps and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield (2014). The

Geometric Sequences Of Discs In The Apollonian Packing

This study began innocently enough with a search for extremal conngurations of circles in the Rodin and Sullivan \Ring Lemma". This is an elementary geometric lemma which nevertheless plays a key

“The Kiss Precise”

IN NATTJBE of June 20 last, we published some verses by Prof. F. Soddy under this title. Shortly afterwards, Mr. E. B. Wedmore sent two verses generalizing Prof. Soddy's equations for the circle and

Introduction to Circle Packing: The Theory of Discrete Analytic Functions

Part I. An Overview of Circle Packing: 1. A circle packing menagerie 2. Circle packings in the wild Part II. Rigidity: Maximal Packings: 3. Preliminaries: topology, combinatorics, and geometry 4.


ANDREEV-THURSTON THEOREM. Let r be a triangulation of S which is not simplicially equivalent to a tetrahedron and let <£: r—»[0, \n\ be given, where T denotes the set of edges of x. Suppose that (A)

Circle Packing: Experiments In Discrete Analytic Function Theory

The fundamentals of an emerging “discrete analytic function theory” are introduced and connections with the classical theory are investigated, ranging from investigation of a conjectured discrete Koebe ¼ theorem to a multigrid method for computing discrete approximations of classical analytic functions.

Minimal surfaces from circle patterns : Geometry from combinatorics

This paper investigates conformai discretizations of surfaces, i.e. discretize parametrized surfaces in terms of circles and spheres, and introduces a new discrete model for minimal surfaces.

Approximation of Quasisymmetries Using Circle Packings

This work develops a means of approximating these quasisymmetries given their associated quasicircles and uses discrete analytic functions induced by circle packings to approximate the Riemann maps.


The paper introduces conformal tilings, wherein tiles have spec- ied conformal shapes. The principal example involves conformally regular pentagons which tile the plane in a pattern generated by a

Fixed points, Koebe uniformization and circle packings

A domain in the Riemann sphere \(\hat{\mathbb{C}}\) is called a circle domain if every connected component of its boundary is either a circle or a point. In 1908, P. Koebe [Ko1] posed the following